Class 13: State Machines

cs2102: Discrete Mathematics
University of Virginia, Fall 2017

See course site for notes:
https://uvacs2102.github.io

Transcript

1. Class 13: State Machines cs2102: Discrete Mathematics | F17 uvacs2102.github.io David Evans | University of Virginia Return Exam 1 Today No problem set due this week! PS6 will be posted soon and due Friday, Oct 20.
2. Plan State Machines Exam 1 Read this week: MCS Chapter 6
3. State Machine
4. Example = , = → , → ) =
5. M =
6. State Machine M = (, : × , ) ∈ ) What does it mean if is total?
7. State Machine M = (, : × , ) ∈ ) What does it mean if is not a function?
8. = = ) = Counter = (, : × , ) ∈ )
9. Is your computer a state machine?
14. http://www.wolframalpha.com/input/?i=2%5E(2%5E41)
15. Modeling as State Machine
16. Modeling as State Machine How accurate is this model? How useful is this model?
18. State Machine Executions
19. State Machine Executions The execution of a state machine, = (, ⊆ ×, ) ∈ ) is a (possibly infinite) sequence of states, () , 3 , … , 5 ) that:
20. Two Steps Forward, One Step Back = ℕ =
21. Two Steps Forward, One Step Back = =
22. Reachable States A state is reachable if it appears in some execution. The execution of a state machine, = (, ⊆ ×, ) ∈ ) is a (possibly infinite) sequence of states, () , 3 , … , 5 ) that: 1. ) = ) (it begins with the start state) 2. ∀ ∈ 0, 1, … , − 1 . > → >?3 ∈ (if and are consecutive states in the sequence, there is an edge → ∈ .
23. = , ∈ ℤ, ∈ {, }} = , → + 1, ∈ ℤ} ∪ , → − 2, ∈ ℤ} ) = (0, ) What states are reachable?
24. = , ∈ ℤ, ∈ {, }} = , → + 1, ∈ ℤ} ∪ , → − 2, ∈ ℤ} ) = (0, )
25. Exam 1
26. 0 20 40 60 80 <80 [80,85) [85,90) [90,95) [95,100) >=100 Exam 1 Histogram
27. Results by Question 0 (Name) 1 (Inference Rules) 2 (Satisfiability) 3 (Logical Equivalence) 4 (Well Ordering) 5 (Relations) 6 (Proofs) 7 (Induction) Average 30.0 9.374 9.887 9.947 8.632 8.726 8.252 8.792 Median 30 10 10 10 8 10 9 10
28. Proofs • To prove by induction, need to state a predicate with an induction variable • Need to write proofs to convince: showing the structure of your argument is more important than showing a bunch of algebra ∷= whatever is here must use n (and that is what you are doing induction on)
29. Returning Exam 1 Group 1 Group 2 Group 3 Group 4 aa8dp (Akanksha) ea6dc (Elmo) kai4vb (Kelsey) qed4wg (Quinn) aap7sc (Alex) eae4bf (Eliza) kc4au (Kunal) rac4ad (Raina) … … … … dz7as (Daniel) jyy3gx (Josh) pw2hm (Peter) zpg9js (Zach) Front of Classroom
30. Returning Exam 1 Group 1 Group 2 Group 3 Group 4 aa8dp (Akanksha) ea6dc (Elmo) kai4vb (Kelsey) qed4wg (Quinn) aap7sc (Alex) eae4bf (Eliza) kc4au (Kunal) rac4ad (Raina) … … … … dz7as (Daniel) jyy3gx (Josh) pw2hm (Peter) zpg9js (Zach) How can we efficiently sort ~80 people by UVA id? Front of Classroom
31. Returning Exam 1 Group 1 Group 2 Group 3 Group 4 aa8dp (Akanksha) ea6dc (Elmo) kai4vb (Kelsey) qed4wg (Quinn) aap7sc (Alex) eae4bf (Eliza) kc4au (Kunal) rac4ad (Raina) … … … … dz7as (Daniel) jyy3gx (Josh) pw2hm (Peter) zpg9js (Zach) Front of Classroom
32. Charge Read MCS Chapter 6 PS6 will be posted soon, due Oct 20